[2608.02478] Solving the Shortest Vector Problem in $2^{0.6039n}$ Time via Mid-point Hessian

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arXiv:2608.02478 (cs)

[Submitted on 3 Aug 2026 (v1), last revised 4 Aug 2026 (this version, v2)]

Title:Solving the Shortest Vector Problem in $2^{0.6039n}$ Time via Mid-point Hessian

Authors:Minki Hhan<br>View a PDF of the paper titled Solving the Shortest Vector Problem in $2^{0.6039n}$ Time via Mid-point Hessian, by Minki Hhan

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Abstract:We present randomized algorithms for the shortest vector problem (SVP). For the $n$-dimensional lattice $\mathcal L$, our algorithms solve SVP in time $2^{0.6039n+o(n)}$ classically and $2^{0.5411n+o(n)}$ quantumly and space $2^{0.5n+o(n)}$, improving the previous best algorithm running in $2^{n+o(n)}$ time and space of Aggarwal, Dadush, Regev, and Stephens-Davidowitz [STOC'15].

Our algorithms heavily use the property of the Hessian of the periodic Gaussian function at the half shortest vector: For a shortest vector $v \in \mathcal L$, the Hessian at $v/2$ has the eigenvector close to $v$, which can be used to recover $v$ using the (preprocessing) bounded distance decoding algorithm. Given the periodicity modulo $\mathcal L$, the candidate midpoints are indexed by the parity classes in $\mathcal L/2\mathcal L$. Our algorithm searches for the class of a shortest vector by estimating the corresponding Hessians using discrete Gaussian samples.

We optimize the algorithm using random sublattice cosets and various sampling technique, achieving the final complexity. The optimization techniques may be of independent interest.

Comments:<br>Corrected title

Subjects:

Data Structures and Algorithms (cs.DS); Cryptography and Security (cs.CR)

Cite as:<br>arXiv:2608.02478 [cs.DS]

(or<br>arXiv:2608.02478v2 [cs.DS] for this version)

https://doi.org/10.48550/arXiv.2608.02478

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arXiv-issued DOI via DataCite

Submission history<br>From: Minki Hhan [view email]<br>[v1]<br>Mon, 3 Aug 2026 16:46:49 UTC (6,884 KB)

[v2]<br>Tue, 4 Aug 2026 03:43:30 UTC (6,885 KB)

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